The second mishna on Chullin 32a explains that if an animal is rendered invalid by any death which does not occur through proper ritual slaughter it is nebelah while if it is rendered invalid for some other reason it is terefah. My question is: what are the consequences of an animal having one of these statuses versus the other? You can't eat it either way, obviously; are there differences in what benefit you can derive from it? What are they, and which is the more severe disqualification?

As the other answers stated, the main difference between נבילה וטריפה is the טומאה. However, there are also other differences, such as אותו ואת בנו (see שמלה חדשה טו); in שלחן ערוך פב where ביצת טריפה is אסור but ביצת נבילה is not; and (lamely) someone who makes a vow נדר not to benefit from one but left the other alone. These are found all over שמלה חדשה for a quick summary see there, 20:6.

One that he does not mention there is also found in Baba Kama (7:2, 7:5), where a thief must pay double כפל for טריפה (because it is a שחיטה) but not for נבילה (because it isn't).

Nebelah: An animal that has died a natural death or was killed not in
accordance with the jewish ritual lawnatural death is called "nebelah"
(carcas). (Bechoros, 3a)

Terefah: An animal afflicted with an organic disease or disability
(eg. the removal of a certain portion of the knee) (Chulin, 42a)

A first-born-Nebelah can have kedushat-Bechor and needs to be redeemed from a Kohen. If it is terefah it loses its Bechor-status and doesn't need redeeming.

According to Rav Hisda, if, when it died, a non-Jew owned any of the parts of the Nebelah that would render the animal treif if that part was missing (eg. windpipe), then the animal is unfit to be a Bechor, and doesn't need to be redeemed. (Bechoros, 3a)